Let us now consider spontaneous transitions between the different energy levels
of a hydrogen atom. Since the perturbing Hamiltonian (1093) does
not contain any spin operators, we can neglect electron spin in our analysis.
Thus, according to Sect. 9.4, the various energy eigenstates of the
hydrogen atom are labeled by the familiar quantum numbers , , and
.

According to Eqs. (1122) and (1131), a hydrogen
atom can only make a spontaneous transition from an energy
state corresponding to the quantum numbers , , to
one corresponding to the quantum numbers , , if
the modulus squared of the associated electric dipole moment

(1144)

is non-zero. Now, we have already seen, in Sect. 12.5, that the
matrix element
is only non-zero
provided that and . It turns out that the proof
that this matrix element is zero unless can, via a trivial modification, also be used to demonstrate
that
and
are also zero unless . Consider

(1145)

It is easily demonstrated that

(1146)

Hence,

(1147)

and

(1148)

Clearly,
is zero unless ,
and
is zero unless .
Now,
and
are
obviously both zero if
and
are both zero. Hence, we conclude that
and
are only non-zero if .

The above arguments demonstrate that spontaneous transitions between different energy levels of a hydrogen atom are only possible provided

(1149)

(1150)

These are termed the selection rules for electric dipole
transitions (i.e., transitions calculated using the electric
dipole approximation). Note, finally, that since the perturbing Hamiltonian
does not contain any spin operators, the spin quantum number
cannot change during a transition. Hence, we have the additional
selection rule that .